Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows

In this work, an efficient gas kinetic scheme is presented for simulation of two-dimensional incompressible thermal flows. In the scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of the Boltzmann equation. To compute these fluxes, two distribution functions are involved. One is the circular function, which is used to calculate the numerical fluxes of mass and momentum equations. Due to the incompressible limit, the circle at the cell interface can be approximately considered to be symmetric so that the expressions for the conservative variables and numerical fluxes at the cell interface can be given explicitly and concisely. Another one is the D2Q4 model, which is utilized to compute the numerical flux of the energy equation. By following the process for derivation of numerical fluxes of mass and momentum equations, the numerical flux of the energy equation can also be given explicitly. The accuracy, efficiency, and stability of the present scheme are validated by simulating several thermal flow problems. Numerical results showed that the present scheme can provide accurate numerical results for incompressible thermal flows at a wide range of Rayleigh numbers with less computational cost than that needed by the thermal lattice Boltzmann flux solver (TLBFS), which has been proven to be more efficient than the thermal lattice Boltzmann method (TLBM).

Figure 1

Distribution of the circular function in the particle velocity space at a cell interface for incompressible flows.

Figure 2

Distribution of D2Q4 model in the particle velocity space at a cell interface.

Figure 3

Error of Nusselt number versus mesh spacing for Rayleigh-Bénard convection at $\mathrm{Ra}=5\times {10}^3$.

Figure 4

The dependence of Nusselt number on Rayleigh number for Rayleigh-Bénard convection.

Figure 5

Streamlines for Rayleigh-Bénard convection at different Rayleigh numbers. (a) $\mathrm{Ra}=5\times {10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}=5\times {10}^4$, (d) $\mathrm{Ra}={10}^5$.

Figure 6

Isotherms for Rayleigh-Bénard convection at different Rayleigh numbers. (a) $\mathrm{Ra}=5\times {10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}=5\times {10}^4$, (d) $\mathrm{Ra}={10}^5$.

Figure 7

Streamlines for natural convection in a square cavity at four different Rayleigh numbers. (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, (d) $\mathrm{Ra}={10}^6$.

Figure 8

Isotherms for natural convection in a square cavity at four different Rayleigh numbers. (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, (d) $\mathrm{Ra}={10}^6$.

Figure 9

Simulation results for natural convection in a square cavity. (a) Streamlines at $\mathrm{Ra}={10}^7$, (b) streamlines at $\mathrm{Ra}={10}^8$, (c) isotherms at $\mathrm{Ra}={10}^7$, (d) isotherms at $\mathrm{Ra}={10}^8$.

Figure 10

Comparison of velocity profiles at the centerlines for natural convection in a square cavity. (a) $\mathrm{Ra}={10}^7$, (b) $\mathrm{Ra}={10}^8$.

Figure 11

Illustration of the setup for natural convection in a concentric annulus.

Figure 12

Streamlines for natural convection in a concentric annulus at different Rayleigh numbers. (a) $\mathrm{Ra}={10}^2$, (b) $\mathrm{Ra}={10}^3$, (c) $\mathrm{Ra}={10}^4$, (d) $\mathrm{Ra}=5\times {10}^4$.

Figure 13

Isotherms for natural convection in a concentric annulus at different Rayleigh numbers. (a) $\mathrm{Ra}={10}^2$, (b) $\mathrm{Ra}={10}^3$, (c) $\mathrm{Ra}={10}^4$, (d) $\mathrm{Ra}=5\times {10}^4$.

Figure 14

Illustration of the setup for mixed heat transfer from a heated circular cylinder.

Figure 15

Streamlines for mixed convection at $\mathrm{Re}=20$ and various Gr. (a) $\mathrm{Gr}=0$, (b) $\mathrm{Gr}=100$, (c) $\mathrm{Gr}=800$, (d) $\mathrm{Gr}=1600$.

Figure 16

Isotherms for mixed convection at $\mathrm{Re}=20$ and various Gr. (a) $\mathrm{Gr}=0$, (b) $\mathrm{Gr}=100$, (c) $\mathrm{Gr}=800$, (d) $\mathrm{Gr}=1600$.